Duality for Finite Gelfand Pairs

Home , Gelfand pair, Zonal spherical function

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1 Introduction 3

2 Duality for group pairs 11 2.1 Dualityinabstractterms ...... 11 2.2 The matrix C(H, G) ...... 15 2.3 Zonalsphericalfunction ...... 17 2.4 Computing matrix C ...... 19 2.5 Arithmetic properties of C ...... 21 2.6 Dualityisomorphism ...... 22 2.7 Galois group of C andduality...... 24 2.8 MapsbetweenGelfandpair ...... 25

3 Examples 27

3.1 (Sn−1,Sn)...... 27

3.2 The group Sn Sk ...... 28 ≀ 3.3 Semidirect product with an abelian group ...... 30 3.4 Nonexamples...... 32 3.5 Onclassiﬁcationofdualpairs ...... 33 3.6 Somesporadicexamples ...... 35

4 Some open questions 35 4.1 On the relation to other dualities ...... 35 4.2 Duality for Gelfand pairs of Lie groups and locally compactgroups ...... 36 4.3 Onthecategoriﬁcation...... 37 4.4 On the suﬃcient conditions for existence of the dual pair ...... 37 4.5 OntheStringTopology ...... 38

2 1 Introduction

Langlands duality for reductive groups and Satake isomorphism are important concepts that are used in the formulation of Langlands conjectures. The author’s motivation for writing this paper is placing these concepts into a more general context with the hope that in the future it might reveal new perspectives on the classical Langlands correspondence. Let us recall (see [8] for an accessible introduction to Langlands theory) some basic defi- nitions needed to state Satake isomorphism. We write G for a split reductive group, K for a non-Archimedean local field, and O for its ring of integers. By following the idea mentioned in the G(O) abstract we are going to replace the linear space Cc[G(K)/G(O))] = Cc[G(O) G(K)/G(O))] \ of Hecke operators by C[H G/H] with finite G and H. As usual, a group in superscript \ V G signals for taking G-invariants. Similarly the space of complexified algebraic characters L [GL(C)]G (C) = [GL(C) GL(C) GL(C)/GL(C)]1 of the Langlands dual group GL(C) over O O \ × C in our setup has a finite analogue C[Hˇ G/ˇ Hˇ ]. In general, we don’t assume that Gˇ splits into \ a product just like GL(C) GL(C). × We would like to think about elements of C[H G/H] C[G] as H-bi invariant functions \ ⊂ on G: f(hgh′) = f(g), g G, h, h′ H. The linear space C[H G/H] has two unital algebra ∈ ∈ \ structures. The first product f g is a point-wise multiplication of functions. The second is · the convolution f g, which in general is noncommutative. Recall that a pair of finite groups × H G is a Gelfand pair iff f g is commutative. Roughly speaking, we want to study duality ⊂ × on Gelfand pairs H G Hˇ Gˇ (1) ⊂ ⇔ ⊂ which interchanges the products . (2) ·⇔× We don’t expect Hˇ Gˇ to exist for an arbitrary Gelfand pair H G. Neither do we hope that ⊂ ⊂ Hˇ Gˇ will be given by a functorial construction. A notable exception is H = 1 and G is ⊂ { } × abelian. Then Hˇ = 1 and Gˇ = HomZ(G, C ). { } 1GL(C) is embedded diagonally into GL(C)) × GL(C)

3 One of the reasons for studying such an amorphous structure (besides formal resemblance with Satake isomorphism) is a link it provides between combinatorics and algebra. We postpone, for now, the formal deﬁnition of duality. Instead we formulate its corollary, which will be proven in due time (see (53)). To state it we notice that the space X = G/H splits into a disjoint union X = oi of H-orbits (in this paper we assume that the map G Aut(X) → has no kernel). TheS space of functions C[X] as a regular G-representation decomposes into a direct sum of irreducible multiplicity-one representations C[X]= i Ti. By a(H, G), b(H, G) we denote the arrays #oi , dim Ti , respectively. One of the nice featuresL of (1) is that it swaps { } { } the sizes of orbits, which live on combinatorial side of the correspondence, and the dimensions of representations, which belong to algebraic side:

a(H, G)= b(H,ˇ Gˇ), b(H, G)= a(H,ˇ Gˇ) and X = Xˇ . (3) | | | |

An illustration It is amusing to see that (H, G) = (Z ,S ) and (H,ˇ Gˇ) = (S ,S S ) is an 4 4 3 3 × 3 example of pairs (H, G) = (H,ˇ Gˇ) which satisfy (3). In fact, it is the simplest example of a non 6∼ self-dual, noncommutative pairs in duality (1). In the second pair, S is embedded diagonally into S S . The orbits of S in S S /S 3 3 × 3 3 3 × 3 3 coincide with the conjugasy classes in S3. These are

1 , (1, 2), (1, 3), (2, 3) , (1, 2, 3), (1, 3, 2) . { } { } { } So a(S ,S S )= 1, 2, 3 . The trivial, sign, and two-dimensional representations exhaust the 3 3 × 3 { } set Irrep(S3) of irreducible representations of S3. By Peter-Weyl isomorphism

C ⊠ [G] ∼= T T . (4) T ∈IrrepM (G) b(S ,S S )= 1, 1, 4 . 3 3 × 3 { } 3 The group of rotations of a cube in R is isomorphic to S4 (it coincides with the group of permutations of diagonals). Let X be the set of faces of the cube. The stabilizer of a face is Z Z Z isomorphic to 4. Thus X ∼= S4/ 4. Under 4 X breaks into a union of two one-point orbits and one four-point orbit. We infer that a(Z ,S )= 1, 1, 4 . Frobenius formula 4 4 { } G G HomH (ResH Ti, C) ∼= HomG(Ti,IndH C) = HomG(Ti, C[X]) = Vi

4 G relates Z4 invariants in Ti with the multiplicities Vi of Ti in C[X]. As usual, IndH C stands G for the G-representation induced from the trivial representation of subgroup H, and ResH T stands for the restriction of T from G to H. Thus all representations Ti that contain the Z4- invariant vectors are subrepresentations of C[X]. Among such is the tautological 3-dimensional representation twisted by sign(σ). In order to construct another subrepresentation of C[X] we notice that S4 can be mapped onto S3. Geometrically this homomorphism comes from the S4 action on pairs of opposite edges of the tetrahedron. Under this map a generator e of Z4 maps to a reﬂection in S3. Because of this, the two-dimensional representation of S3 pulled back to

S4 contains a Z4-invariant vector. Constants deﬁne a trivial subrepresentation in C[X]. Note that dim C + dim C2 + dim C3 = dim C[X] = 6. We conclude that b(Z ,S )= 1, 2, 3 . We see 4 4 { } that a(S ,S S )= b(Z ,S ), b(S ,S S )= a(Z ,S ). 3 3 × 3 4 4 3 3 × 3 4 4 For more examples the reader can consult Section 3.

Algebras with two multiplicative structures Linear spaces C[H G/H] have more struc- \ tures then multiplications and mentioned above. We are going to enhance C[H G/H] by · × \ these structures and put the resulting object inside of a certain category. This will let us state in more precise terms what it means to swap the products in formula (2). We will be writing X for G/H. We start with an observation that

the space C[X X]G is isomorphic to C[X]H = C[H G/H] : (5) × ∼ \ r : f(x,x′) f(x,x′ ), with St(x ) equal to H. The two products , on C[H G/H] and other → 0 0 × · \ structures are easier to describe in terms of C[X X]G. This what we are going to do now. × To this end, we start in a greater generality by ﬁxing ﬁnite sets X,Y , which at the moment have no relation to the pair (H, G). C[X Y ] is an algebra with respect to the point-wise × multiplication of functions

(f g)(x,y) := f(x,y)g(x,y) with the unit 1 (x,y) = 1. (6) · ·

5 The convolution product C[X Y ] C[Y Z] C[X Z] is given by × × × → × (f g)(x,y) := f(x,y)g(y, z), f C[X Y ], g C[Y Z]. (7) × ∈ × ∈ × y X X∈ C[X Y ] is isomorphic to the space of linear maps HomC(C[Y ], C[X]). The isomorphism is × deﬁned by the formula

(f h)(x) := f(x,y)h(y), h C[Y ]. (8) × ∈ y Y X∈ Under this identiﬁcation, the convolution becomes a composition of maps. In particular, C[X × X] is an algebra with unit 1 (x,y) = δxyx,y X and C[X Y ] is a C[X X], C[Y Y ] × ∈ × × × bimodule. C[X X] is equipped with two functionals: ×

tr·(f) := f(x,y) tr×(f) := f(x,x) (9) x X (x,y)X∈X×X X∈

(a, b) := tr (a πb)=tr (a µb) (10) · · × × and is positive-deﬁnite: (a, a) > 0, a = 0. (11) 6 The units satisfy 1 1 = X 1 1 1 = 1 (12) · × · | | · × · × × We are going to use to complex anti-linear involutions on C[X X]. The ﬁrst one is the complex × conjugation map π(f)(x,y)= f¯(x,y). (13)

The second is a Hermitian conjugation

(µf)(y,x) := f¯(x,y). (14)

Note that πµ = µπ (15) and

tr·(πa)= tr·(a), tr·(µa)= tr·(a), (16) tr×(πa)= tr×(a), tr×(µa)= tr×(a).

6 Traces are related by the identities

tr (f)=tr (f 1 ), tr (f)=tr (f 1 ). (17) · × × · × · · × Suppose now that X = Y = Z = G/H for the ﬁnite H G. All of the above structures ⊂ are compatible with the regular left G action on C[X X]. This way C[X X]G inherits the × × products and , the traces tr and tr , and the involutions π and µ. · × · × Our basic objects will be Frobenius algebras with some additional data:

U := (C[X X]G, , tr (1 )−1tr ,π ,µ ) × · · · · · ·

π· := π,µ· := µ and

W := (C[X X]G, , tr (1 )−1tr ,π ,µ ) × × × × × × ×

π× = µ,µ× = π

There is a tautological pairing between linear spaces U and W

a, b := tr·(a π(b)), a U, b W h i · ∈ ∈ (18) which satisﬁes a, µ b = π a, b , a, π b = µ a, b h × i h · i h × i h · i To summarize, we have a construction

H, G (H, G) ⇒A where is the triple (U, W, , ). (19) A h· ·i Deﬁne a new triple ˇ as (W, U, , ) by interchanging U and W (c.f. this with the deﬁnition A h· ·i from Section 3 in [6]).

Remark 1 We want to think about triples (W, U, , ) as objects of a category . In we h· ·i T T don’t insist on trU and trW being nonzero, nor on dim U and dim W being ﬁnite. Still we require , to be a nondegenerate pairing. The morphisms in are homomorphisms (inclusions) of h· ·i T underlying algebras compatible with inner products involutions and traces. is a monoidal T category with respect to the tensor product of triples.

7 We would like to explore the following problem: Under what conditions on the pair (H, G) is there a dual pair (H,ˇ Gˇ), such that there is an isomorphism in T

D : (H, G) = ˇ(H,ˇ Gˇ), (20) A ∼ A and what is the freedom in the choice of (H,ˇ Gˇ)? Notice that if we formally replace H, G in (20) by G(O), G(K) and H,ˇ Gˇ by GL(C), GL(C) × GL(C) the classical Satake isomorphism identifies W (H, G) and U(H,ˇ Gˇ). It is part of the structures needed for (20) to hold. Of course, in the case of infinite groups one has to be careful about analytic aspects, which will be ignored in our finite setting.

Remark 2 The algebra C[H G/H] acts on C[G/H] from the right by endomorphisms of G- \ regular representation in C[G/H]. This observation is used in the proof of the criteria Lemma 3.9. p.42 [10]: (H, G) is a Gelfand pair ⇔

C[G/H]= Ti Vi(G/H), Ti = Tj, dim Vi 1. (21) ⊗ 6 ≤ i M In this formula Ti stands for irreducible G-representations, and dim Vi(G/H) for their multiplicities.

The most well-known example of a Gelfand pair is G embedded diagonally into G G. This × follows, for example, from Remark 2 and Peter-Weyl theorem [14].

Relation to topology It turns out that structures similar to (H, G) appear in topology. A To see this, we fix a connected (for simplicity) oriented submanifold N in a finite-dimensional compact connected manifold M. Let us define the space of paths

L(N, M) := γ : [0, 1] M γ(0), γ(1) N . { → | ∈ }

The ﬁrst indication that L(N, M) has some relation to previous constructions is that the set of connected components π0(L(N, M)) is isomorphic to i(π (N)) π (M)/i(π (N)) (see e.g. [2] p.397 Corollary 10.7.6). The map i stands for embedding 1 \ 1 1

8 N M. In order to simplify the notations we will often omit i and the base point in the → formulas. The space C[π (L(N, M))] = C[π (N) π (M)/π (N)] is isomorphic to zero cohomology 0 1 \ 1 1 0 H (L(N, M), C) and zero homology H0(L(N, M), C). ∗ i These linear spaces are parts of a richer structure. The direct sum H (N, M)= i≥0 H (L(N, M), C) is the graded commutative algebra with respect to the -product. It has twoL involutions ∪ π(a) =a ¯. Involution µ is a composition of complex conjugation π and geometric operation or∗ that changes orientation of a path or(γ)(t)= γ(1 t). − Homology groups H∗(N, M) := i≥0 Hi(L(N, M)) has an algebra structure L Hi Hj Hi j N ⊗ → + −dim defined by concatenation of paths (see [15],[9]). This product is denoted by a b. • By definition a developable orbifold M is quotient of a manifold M by the discrete group Γ acting properly. In this case π (M) by definition is equal to Γ. In our application we choose N 1 f to be a Γ-invariant submanifold. e Example 3 To make connection with the algebraic discussion we take M to be a compact simply connected manifold equipped with a finite group G action. For N we take a G-orbit of a point f y M. Denote by p : M M the canonical projection. The submanifold N M is an orbifold ∈ → e ⊂ point x = p(N). In this setup π (M) := G and π (N) := St(y)= H. Now N is disconnected f{ } f 1 1 but the appropriate modifications of the previous constructions go through. e e The product on H (N, M) extends without troubles to developable orbifolds (see [11] for M ∗ ⊂ M M case). The linear space H ( x , M) with the product is the Hecke algebra of the pair of × 0 { } × groups (St(y), G). The space H0( x , M) with -product coincides with (C[St(y) G/St(y)], ). { } ∪ \ · Now we would like to assume that N is an even-dimensional manifold. We reduce the grading ∗ in algebras H∗(N, M), H (N, M) modulo two. This way we get a triple

(N, M) = (U, W, , ), B h· ·i U = (H∗(N, M), , tr = 0,π ,µ ),π := π,µ = µ, (22) ∪ ∪ ∪ ∪ ∪ W = (H (N, M), , tr = 0,π ,µ ),π = µ,µ = π. ∗ • • • • •

9 In the above formulas , stands for the pairing between homology and cohomology. Had h· ·i L(N, M) been a finite-dimensional manifold, we could have identified homology with cohomology by the Poincare duality pairing. That would have given us a triple similar to the one appearing A in the group case. In reality, L(N, M) is a infinite-dimensional space. Thus the triple is a B substitute for in infinite dimensions. There is a duality operation acting on a triple which A B interchange H (N, M) and H∗(N, M). Denote the result of such modification by ˇ. ∗ B We say that a pair of manifolds Nˇ Mˇ is dual to N M if ⊂ ⊂ (N, M) = ˇ(N,ˇ Mˇ ). B ∼ B

As in the case with groups, in order for this isomorphism to hold, H∗(N, M) must be a commutative algebra. In this case we will refer to the pair (N, M) as a topological Gelfand pair. As in the group case, the diagonal embedding M M M of orientable M gives an example → × a topological Gelfand pair. Indeed the space L(M, M M) is homeomorphic to the free loop × space L(M) = Maps(S1, M). (A pair of restrictions of γ L(M) on semicircles S1 S1 = S1 ∈ + ∪ − gives an element in L(M, M M).) The product on H (M, M M) coincides with the × × ∗ × • product H∗(L(M)), which is commutative (see [3] for details). It is not obvious that the proposed duality is nontrivial. In the following sections we will see that there are plenty of examples of dual group pairs. Here is an outline of the paper. The text is divided roughly in two parts: theoretical material (Section 2) and a collection of examples (Section 3). Here is a more detailed breakdown. In Section 2 we study at some length the general aspects of the duality. Thus in Section 2.1 we isolate features of duality that can be formulated without mentioning Gelfand pairs. In Section 2.2 we discuss properties of the duality related to the combinatorics of G-action on G/H. Section 2.3 contains, besides the well-known material about spherical functions, the formulas for idempotents for -product. The general formula for matrix C is derived in Section × 2.4. Some algebraic and arithmetic properties of the coeﬃcients of matrix C are determined in Section 2.5. Based on this we formulated a necessary condition for existence of the dual pair H,ˇ Gˇ in Section 2.6. Conjectural relation of the Galois group of the splitting ﬁeld and duality is discussed in Section 2.7. In Section 2.8 we set up a language for studying (non)uniqueness

10 of the dual pair. Section 3 is devoted to examples of dual pairs described with various levels of precision. In Sections 3.1, 3.2, and 3.3 we manually compute the triples (A,B,C). In Section 3.4 we illustrate how a necessary condition established in previous sections work in the simplest example. Rudiments of computed-aided classiﬁcation of dual pairs are presented in Section 3.5. In Section 4 we listed some of the open problems in the outlined theory.

Acknowledgment The author beneﬁted from conversations on the subject of this paper with P. Bressler, P. Deligne, S. Donaldson, O. Gabber, D. Kaledin, M. Kontsevich, A. Kuznetsov, L. Mason, R. Matveyev, A. Mikhailov, A.S. Schwarz, D. Sullivan, L. Takhtajan, O.Viro. The author owes special thanks to P. Terwilliger who has pointed at [6]. That work contains much overlap with Section 2.1. Part of the work on this project was conducted at Mathematical Institute University of Oxford, ICTP (S˜ao Paulo), and IHES. The author would like to thank these institutions for excellent working conditions.

2 Duality for group pairs

We will see in this section that a triple (H, G) (19) can be eﬀectively encoded by a square A matrix C(H, G). In terms of this matrix, the duality corresponds to taking the Hermitian adjoint C(H, G)†.

2.1 Duality in abstract terms

The goal of this section is to give a more economic description for the triple (19). For this purpose we take a linear space Q as a prototype for C[H G/H]. We assume that Q is equipped \ with two commutative algebra structures and , has two traces tr and tr , has two commuting · × · × (15) anti-linear involution π,µ compatible with both multiplications. We require that traces are real (16). They also satisfy (10),(11). Note that equations (17) are formal corollaries of (10) and (15).

11 We demand that units satisfy (12). Denote by (Q) the triple (U(Q),W (Q), a, b ) where A h i U(Q) := (Q, , tr (1 )−1tr ,π,µ), · · · · W (Q) := (Q, , tr (1 )−1tr ,µ,π), (23) × × × × a, b = (a, b), a U, b W. h i ∈ ∈ We ﬁx notations for normalized traces:

−1 −1 TrU := tr·(1·) tr·, TrW := tr×(1×) tr×.

Deﬁnition 4 [6] The septuple (Q, , , tr , tr ,π,µ) is called a character algebra (c.f. Deﬁnition · × · × 2.1 loc.cit).

Proposition 5 Under above assumption the triple (Q) (23) determines a character algebra A (Q, , , tr , tr ,π,µ). · × · ×

Proof. The only diﬀerence between (Q) and (Q, , , tr , tr ,π,µ) is that of normalization of A · × · × traces in (Q). As soon as we recover the normalization constants tr (1 ), tr (1 ) we recover A × × · · the septuple.

From nondegeneracy of TrU , TrW (11) we derive existence of elements eU U, eW W such ∈ ∈ that

a, 1 = TrU (a eU ), a U, 1 , b = TrW (eW b), b W. h ×i · ∈ h · i × ∈

It follows from (17) that eU = tr (1 )1 U and eW = tr (1 )1 W . The (Q)-dependent · · × ∈ × × · ∈ A constants K,N are deﬁned by the formulas

K := eU , 1 = 1 , eW = tr (1 )tr (1 ), h ×i h · i · · × × (10) N := 1 , 1 = tr (1 )=tr (1 µ1 ) = tr (1 π1 )=tr (1 ). h · ×i × · × · × × · · · × · ×

Units 1·, 1× are obviously invariant with respect to π and µ. We have the following (in)equalities:

(11) (10) (12) 0 < tr (1 )=tr (1 π1 ) = tr (1 µ1 ) = X tr (1 )= X N, · · · · · · × · × · | | × · | |

12 (11) (10) (12) 0 < tr (1 )=tr (1 µ1 ) = tr (1 π1 ) = tr (1 )= N. × × × × × × · × · × · × We conclude that

tr (1 )= K/N, tr (1 )= N, tr (1 )=tr (1 )= N, X = K/N 2. · · × × · × × · | |

Proposition 6 1. In a character algebra E = (Q, , , tr , tr ,π,µ) (Q, ) and (Q, ) are · × · × · × isomorphic to direct products Cdim Q of ﬁelds.

2. We ﬁx an inner product as in (10) and two sets A, B, A = B = dim Q. Up to an | | | | isomorphism E is completely characterized by two real vectors A CA,B CB and a ∈ ∈ matrix C such that

(Xi, Xj)= Aiδij, Ai = tr·Xi, Ai > 0, (24)

(Ψi, Ψj)= Biδij,Bi = tr·Ψi,Bi > 0, (25)

(Xi, Ψj)= Cij. (26)

Xi i A is the basis of minimal idempotents deﬁned with respect to multiplication . { | ∈ } · Ψj j B is the similar basis for -multiplication. There are involutions µ : A A, { | ∈ } × → π : B B such that →

Ai = Aµ(i),Bi = Bπ(i),Ciπ(j) = Cij,Cµ(i)j . = Cij (27)

The isomorphism of two triples (A,B,C), (A′,B′,C′) is a pair of permutations σ, τ of indices A, B such that A = A′ ,B = B′ ,C = C′ , i A, j B. σ intertwines σ(i) i τ(j) j σ(i)τ(j) ij ∈ ∈ µ and µ′. τ intertwines π and π′.

The data satisfy dim Q −1 Bkδkl = Ai CikCil. (28) i X=1 3. By (12) 1 / X is an idempotent for -multiplication, 1 - for -multiplication. Thus · | | × × ·

1 / X = niΨi, 1 = miXi,ni,mi = 0, 1. · | | × i i X X 13 Then

Ai = X Cijnj, Bj = Cijmi. (29) | | j j X X In case 1 / X = X , 1 =Ψ are minimal idempotents, (30) · | | 1 × 1 then

Ai = X Ci , Bj = C j | | 1 1 (31) trA = X Ci , trB = C j. | | 1 1 i i X X Proof. Algebras (Q, ) and (Q, ) are semisimple. Indeed both π,µ permute maximal ideals m · × { } of (Q, ) and act on their intersection rad = m. They also preserve filtration rad rad2 · · · ⊃ · ⊃ k . Let rad be the last nonzero term of thisT filtration (we use that rad· is nilpotent). Then · · · · (a, a) =tr (a πa) = 0 rad = 0 . Thus (Q, ) is a direct sum of fields. π permutes minimal · · ⇒ · { } · idempotents Xi i A . We have 0 < tr (Xi πXi). From this we conclude that π acts trivially { | ∈ } · · on Xi . { } We can prove the same way that (Q, ) is semisimple and µ acts trivially on the set of × minimal idempotents Ψi i B . We have tr (Xi πXj)=tr (Xi Xj)=tr (Xi)δij = δijAi. From { | ∈ } · · · · · positivity of ( , ) we conclude that Ai are positive reals. The proof that µΨi =Ψi and Bi > 0 · · { } is similar.

Operator µ deﬁnes an involution on A: Xµ(i) := µXi. The same way π deﬁnes an involution on B: Ψπ(i) := πΨi. By abuse of notation we denote the involution of the set of idempotents by the symbol of the automorphism it induces.

Let us expand Ψi in the basis of Xi:

k Ψj = dsjXs. (32) s=1 X Then

Cij = (Xi, Ψj)= dijAi. (33)

14 Equations (25) and (32) imply that

k k −1 Biδij = (Ψi, Ψj)= dsidsjAs = As CsiCsj. s s X=1 X=1 −1 −1 −1 It is equivalent to the statement that (C )ij = Bi CjiAj , which implies that

k −1 Aiδij = Bs CisCjs. (34) s=1 X We leave to the reader the proof of (27) which uses πµ = µπ and (16). The triple (A,B,C) is sufficient to recover two algebra structures up to an isomorphism. dim Q For this purpose we choose the standard basis Xi for the vector space C . We declare it { } a basis of idempotents for -multiplication, tr (Xi) := Ai, µXi := X , πXi := Xi. π and µ · · µ(i) obviously commute. Positivity of tr (a πb) follows from (24). · · We define the basis Ψi of idempotents for multiplication by the formula (32), where the { } × matrix d is extracted from (33). We also define tr× by tr×(Ψi) := Bi. The formula (10) would follow from (34). Equations (29),(31) follow from

Remark 7 Tensor product in monoidal category transforms into tensor product of matrices T C,C′ C C′ and direct product of involutions π,π′ π π′, µ,µ′ µ µ′. Recall (see → ⊗ → × → × Proposition 6) that involutions in the context of matrix C are acting on indices of C.

2.2 The matrix C(H,G)

When (H, G) is a Gelfand pair, collection (Q(H, G) = C[X X]G, , , tr , tr ,π,µ) (with the × · × · × structures deﬁned in (6,7,9,13,14)) deﬁnes a septuple E(H, G) and a triple (Q(H, G)) from A

15 Proposition 5. Semi-simple multiplications define bases Xi and Ψi of minimal idempotents { } { } in Q(H, G). This way we get a triple (A,B,C) (24,25,26) associated with (H, G). By Propositions 5 and 6 the triple (H, G) contains as much information as the triple A (A,B,C) and involutions π,µ. (27). We will see that (30) is satisfied and C completely determines A and B. Our goal for now is to find explicit formulas for (A,B,C) that come from a finite Gelfand pair H G. ⊂ From now on to the end of the paper we assume that X = G/H.

Remark 8 Let Oi X X be a G-orbit in X X so that ⊂ × ×

X X = Oi. (35) × i G Oi determines an H orbit oi X: ⊂

x oi := Oi X x ,H = St(x ). { 0}× ∩ × { 0} 0

G Construction of Xi C[X X] is obvious: Xi is the characteristic functions of Oi. The ∈ × Gram-Schmidt of Xi is { }

(Xi, Xj)= Oi δij = δij X oi = δijAi. (36) | | | || |

Remark 9 Note that 1 = X that corresponds to the diagonal X X X is a minimal × 1 ⊂ × -idempotent. ·

k G Proposition 10 1. In the product of -idempotents Xi Xj = aijsXs, Xi C[X X] · × s=1 ∈ × P the coeﬃcients aijs are nonnegative integers. (37)

2. The integrality condition (37) in the coordinate form becomes

−2 −1 ≥0 CisCjsBs CmsAm Z . (38) s ∈ X Proof. Suppose (x,y) Os, where Os is the orbit that deﬁnes Xs. Then aijs is the number of ∈ z X such that (x, z) Oi, (z,y) Oj. ∈ ∈ ∈ We leave veriﬁcation of the second statement to the reader.

16 Corollary 11 We can use the triple (A,B,C) to write transition matrices between bases Xs { } and Ψj : { } k Ψj = Csj/AsXs s=1 X (39) k Xs = Csj/BjΨj. j X=1 2.3 Zonal spherical function

The basis Xi was identiﬁed in Remark 8. The description of the basis Ψj uses the concept { } { } of zonal spherical function, or spherical function for short.

Definition 12 Let (T (g),V ) be a unitary finite-dimensional representation of of a finite G. We associate a function on G defined by the formula: dim T ψ (g)= (T (g)θ,θ) (40) θ X | | with a unit H-invariant vector θ. We call ψθ(g) the spherical function associated with a subgroup H and unit invariant vector θ.

′ The function ψθ(g) satisfies ψθ(hgh )= ψθ(g) and descends to a function on H X. \ We will be writing SpecG(C[X]) for collection of irreducible representations with no rep- etitions that appear in (21). Each representation is equipped with a fixed positive-definite

Hermitian inner product. Let Ti be a representation from SpecG(C[X]). By construction G C[X]= IndH C. By Frobenius formula

G G HomH (ResH Ti, C) ∼= HomG(Ti,IndH C) = HomG(Ti, C[X]) = Vi ∼= C.

We conclude that H-invariant vector θi in Ti is unique up to a scaling factor. We normalize it

(θi,θi) = 1.

Convention 13 It is convenient to use the elements of the set SpecG(C[X]) as labels of functions ψ(g). Thus ψi(g), i SpecG(C[X]) will stand for the spherical function ψθ(g), where θ is ∈ a normalized H-invariant vector in the representation T which belongs to the isomorphism class i.

17 ¯′ ¯′ We deﬁne a Hermitian inner product on G by the formula (f, f )= g∈G f(g)f (g). P Proposition 14 Functions ψi(g) are orthogonal with respect to the inner product in C[G] { } and deﬁne a basis in C[X]H .

Proof. We extend vector θi = θi, to an orthonormal basis θi, ,...,θ . The function 1 { 1 i,dim(Ti)} ψi(g) is the matrix coefficient Ti,1,1(g) of the representation Ti. The following identity is a simple corollary of the Schur orthogonality relation for matrix coefficients of finite groups:

By orthogonality ψi are linearly independent. Their number coincides with cardinality SpecG(C[X]) , | | which is equal to the number of summands in (21). By assumption dim(Vi(X)) 1. Thus, by ≤ (5) and Maschke isomorphism

(8) (21) k G HomG(C[X], C[X]) = C[X X] = HomC(Vi,Vi) (42) ∼ × ∼ i M=1 G H SpecG(C[X]) = dim C[X X] = dim C[X] . | | × Corollary 15 We interpret H H invariant function ×

−1 ′ Ψi = ψi(g g ) (43) on G G as a function on G/H G/H = X X. By construction Ψi is invariant under × × × diagonal G-action. Orthogonality relation

Bij = Ψi(x,y)Ψj(x,y)= δi,j dim Ti (44) x,y X X ∈X× follows from (41).

Proposition 16 The functions Ψi are minimal idempotents in (C[X X], ). The function × × Ψ corresponding to trivial representation is equal to 1 / X . 1 · | |

18 Proof. Schur orthogonality relations yield the proof: 1 Ψ (x,y)Ψ (y, z)= ψ (g−1g′)ψ (g′−1g′′)= i j H i j y X ′ X∈ | | gX∈G 1 dim Ti dim Tj −1 ′ ′−1 ′′ 2 Ti,1,s(g )Ti,s,1(g )Tj,1,t(g )Tj,t,1(g )= H X ′ | | | | gX∈G 1 dim Ti dim Tj −1 ′ ′ ′′ 2 Ti,1,s(g )Ti,s,1(g )T j,t,1(g )Tj,t,1(g )= H X ′ | | | | gX∈G 2 1 dim Ti G −1 ′′ −1 ′′ ′′ δi,j 2 | | Ti,1,1(g g )= δi,jψi(g g )= δi,jΨi(x, z), x = gH,z = g H H X dim Tj | | | |

2.4 Computing matrix C

Having obtained explicit formulas for Xi, Ψj we can compute matrix C (matrices A and B were computed in (36) and (44)).

Proposition 17 In each H-orbit oi X choose ai oi. The indexing is chosen so that ⊂ ∈ St(a )= H. We also ﬁx gi G such that ai = gia . There is a bijection between gi and Xi . 1 ∈ 1 { } { } In the notations of isomorphism (21)

Cij = (Xi, Ψj)= oi dim Tj(θj, Tj(gi)θj) (45) | |

θj is a normalized H-invariant vector in Tj.

Proof.

19 Proposition 18 Matrix C satisﬁes

Cij = δ j X , (47) 1 | | i X Cij = δi X (48) 1| | j X ′ 1 ′ Proof. The ﬁrst equality is the corollary of orthogonality of Ψ (x,x )= and Ψi(x,x ) i = 1 |X| { | 6 1 . Indeed, by taking Oi as in (35) and using the ﬁrst line of (46) we get } 1 1 1 δ = Ψ (x,x′)= Ψ (x,x′)= C . 1,j X j X j X i,j ′ i ′ i | | x,xX∈X | | X (x,xX)∈Oi | | X This is equivalent to (47).

The second equality follows from Fourier expansion of the characteristic function ∆X of the diagonal X X X : ⊂ × 1 ∆ (x,x′)= c Ψ (x,x′), c = ∆ (x,x′)Ψ (x,x′)= X j j j dim T X j j j ′ X x,xX∈X X = | | ψj(1) = 1. dim Tj ′ ′ So j Ψj(x,x )=∆X (x,x ). In terms of the functions ψj and characteristic function δH (g) of theP subgroup H this equality becomes

dim Tj(Tj(g)θj ,θj)= X δH (g). | | j X On substituting g gi and multiplying both sides by oi this equality becomes equation (48). → | |

Corollary 19 Matrix C completely determines diagonal matrices A and B

C i = Ci = X Ai/ X = Ci , Bi = C i (49) 1 1 | | | | 1 1 i i X X This follows from the inspection of (46) or from Proposition 6 item (3), Remark 9, and Propo- sition 16.

Remark 20 Matrix C for (G, G G) (G is embedded diagonally) is nothing else but the × character table for the group G.

20 2.5 Arithmetic properties of C

In this section we will take a closer look at the coeﬃcients of the matrix C, which was derived from a Gelfand pair (H, G).

Proposition 21 The coeﬃcients of C generate a Galois extension L of Q whose Galois group is abelian. L is a minimal splitting ﬁeld for regular representation G in Q[G/H].

Proof. Recall that exponent e(G) of a finite group G is a minimal positive number such that ge(G) = 1 g G. By [5] Theorem 41.1 p. 292 any irreducible representation T of a finite ∀ ∈ k group G defined over K = Q[√1], k = e(G) has HomG(T, T ) = K. The convolution algebra K[X, X]G, being an algebra of endomorphisms of K[X], by the above theorem is a direct sum n G G K (see (42) with C replaced by K). Basis Xi belongs to Q[X, X] K[X, X] . By the i=1 { } ⊂ citedQ theorem idempotents Ψi are defined over K and the inner products (Xi, Ψj) K. Let { } ∈ × L be a subfield of K generated by (Xi, Ψj). The Galois group of K is an abelian group Zk . By the elementary Galois theory there is a subgroup H(L) Z×, which fixes elements of L and ⊂ k × Gal(L/Q)= Zk /H(L).

Q[G/H] splits over L because of the formula (39) for splitting idempotents Ψj.

Proposition 22 The entries of the matrix Csj/Bj are algebraic integers.

Proof. Let us expand Xs in the basis Ψj as in (39): By deﬁnition Ψj is an idempotent for { } -product. From this we deduce that ×

Xs Ψj = Csj/BjΨj ×

It means that Ψj is an eigenbasis for the operator Lsy := Xs y and Csj/Bj are its eigenvalues. { } × The matrix of Ls in the basis Xs has integer coefficients (Proposition 10). The characteristic { } polynomial of this matrix is monic with integer coefficients. By definition its roots are algebraic integers.

Remark 23 By Proposition 10.2 Section I in [13] p.60 the set of algebraic integers in Q[√k 1] (k = e(G)) is a free module Z[ζ] over Z. As a ring the module Z[ζ] is generated by a primitive

21 root of unity ζ. Let Wk Z[ζ] be the group of roots of unity. Wk contains a subset Ak such ⊂ × × that Ak = Z (the orbit of ζ under the action of Galois group Z ) such that Ak is a basis for | | | k | k Z[ζ]. By utilizing Corollary 19 we get

r Csj/C1j = msjrζ ,msjr Z. (50) r ∈ ζX∈Ak 2.6 Duality isomorphism

Let us spell out explicitly how duality D (20) isomorphism is deﬁned. Let (H, G) be a Gelfand pair and (H,ˇ Gˇ) be the dual Gelfand pair.

We choose a basis of minimal idempotents Xi for U and Ψj for W . There are similar { } { } bases Xˇ i for Uˇ and Ψˇ j for Wˇ . The duality map D acts by { } { } ˇ ˇ D(Xi)= Ψσ(i), D(Ψj)= Xτ(j)

σ and τ are some bijection between indexing sets. The fact that D is an isomorphism of structures (see Remark 1) manifests itself in equalities

ˇ Cij = Cτ(j)σ(i). (51)

Now it will be convenient to identify the set of labels A, B with the set 1,..., X in a such a { | |} way that Ai Ai ,Bi Bi . Without loss of generality we can assume that τ = σ = id. It ≤ +1 ≤ +1 follows from Corollary 19 that

oi = dim Tˇi. (53) | |

22 It means that duality interchanges sizes of H-orbits a = A/ X and dimensions b = B of | | irreducible components the regular G representation in C[X]. We have seen this in the example 1. Duality transformation produces from the triple (A,B,C) a new triple

−1 † (Aˇ := X B, Bˇ := X A, Cˇ := C ) where X := Bi (54) | | | | | | i X withπ ˇ = µ, µˇ = π as in (27). In order for the triple (A,ˇ B,ˇ Cˇ) to arise from a Gelfand pair (H,ˇ Gˇ) it must satisfy some consistency conditions.

Proposition 24 The triple (A,ˇ B,ˇ Cˇ) satisﬁes (28),(47),(48),(49) with all the variables replaced by their checked modiﬁcations.

−1 −1 Proof. Equation (28) tells us that Aii CijBjj is the inverse to the transpose of Cij. From this −1 −1 we infer that k CikAjj CjkBkk = δij, which is equivalent to P 1 1 CikCjk = Ajjδij. X Bkk X Xk | | | | Duality interchanges equations (47),(48). It also swaps last pair of equations in (49).

Deﬁnition 25 (A,B,C) is called an integral triple if (A,ˇ B,ˇ Cˇ) satisﬁes (38) and (50). In terms of (A,B,C) this conditions reads

−2 −1 ≥0 X CsiCsjAs CsmBm Z i, j, m (55) | | s ∈ ∀ X r Cis/Ci1 = nisrζ ,nisr Z, k = e(G) i, s. (56) r ∈ ∀ ζX∈Ak Strictly speaking the second equation is a corollary of the ﬁrst. Still, it is worthwhile to write it separately because it is easy to verify.

Deﬁnition 26 (A,B,C) is called a self-dual triple if Ai = Bi and there are σ, τ such that

Ai = Aσ(i), Bi = Bτ(i) and Cji = Cσ(i)τ(j). Moreover σπ = µσ, τπ = µτ.

23 Deﬁnition 27 Let L be a minimal splitting extension of Q for regular representation Q[X]. An element (σ, τ, g) S S Gal(L/Q) is a symmetry of the triple (A,B,C) if Ai = A , ∈ |X| × |X| × σ(i) ab Bi = B , Cij = gC . σπ = πσ, τµ = µτ. As L Q C, g commutes with the complex τ(i) σ(i)τ(j) ⊂ ⊂ conjugation.

2.7 Galois group of C and duality

Let L be the splitting ﬁeld of the regular Gˇ-representation Q[Xˇ]. An element g of the Galois group Gal(L/Q) (which we know is abelian) permutes irreducible sub-representations of L[Xˇ].

This way g defines a permutationτ ˇg of idempotents Ψi . As a result, we have an element { } ˇ ˇ ˇ (1, τˇg−1 , g) of the group Aut(A, B, C) defined in (27). By abuse of notation, we will say that † (1, τˇg, g) is an element of Gal(L/Q). The coefficients of C and C generate the same subfield of

C. From this we conclude that (ˇτg, 1, g) is a symmetry of (A,B,C).

Let us define groups AutH (G) := φ Aut(G) φH H and OutH (G) := AutH (G)/H. { ∈ | ⊂ } An element k of AutH (G) defines a permutation αk of double cosets H G/H. Homomorphism \ k αk factors through OutH (G). We interpret αk as a permutation of idempotents Xi . → { } AutH (G) defines an automorphism of the group algebra L[G]. It leaves invariant subalgebra

C[H G/H]. This way we get a permutation βk of idempotents Ψi . To summarize: we have a \ { } homomorphism ν : OutH (G) S S Gal(L/Q), ν(k) = (αk, βk, 1), which is a symmetry → |X| × |X| × of (A,B,C) related to H, G. We will refer to elements (αk, βk, 1) k Z(OutH (G)) as elements { | ∈ } of ZH (G).

In the above setup we conjecture that (ˇτg,τg−1 , 1) is an isomorphism Gal(L/Q) with ZH (G). Gal(L/Q) is the same for H, G and H,ˇ Gˇ. Conjecture implies that

ˇ ZHˇ (G) ∼= ZH (G).

Suppose that Aut(G) = G/Z(G). In this case OutH (G) = N(H)/H. The group N(H) is the normalizer of H in G. There is an embedding C[N(H)/H] C[H G/H]. So if C[H G/H] is → \ \ -commutative then N(H)/H must be commutative. Under above assumptions we conjecture × that Gal(L/Q) ∼= N(H)/H.

24 2.8 Maps between Gelfand pair

A map (H, G) (H′, G′) of Gelfand pairs is a homomorphism φ : G G′ that maps H to H′. → → A map φ is a strong Hecke equivalence if it induces a bijection φ : G/H G′/H′. We say that → φ is a Hecke equivalence if it induces a bijection H G/H H′ G′/H′. \ → \ Proposition 28 Hecke equivalence φ : (H, G) (H′, G′) of ﬁnite Gelfand pairs implies strong → Hecke equivalence.

Proof. φ : G/H =: X X′ := G′/H′ induces a map of representations φ∗ : C[X′] C[X] via → → pullback. By the assumption it deﬁnes an isomorphism of invariants

′ C[X′]H C[X]H . (57) → ′ By Proposition 14 spherical functions ψ′ form a basis in C[X′]H . In addition the span { i} of left shifts ψ′(gx),x X′ of ψ′(x) coincides with the irreducible representation T ′. The i ∈ i i ′ similar construction applied to ψi(φ(x)) generates an isomorphic subrepresentation in C[X]. ′ ′ Representations Ti come with multiplicity one and generate C[X ]. From this we conclude ∗ ′ that φ is injective. From isomorphism (57) we see that ψj(x) = aijψi(φ(x)) where (aij) is an invertible matrix. We know that spherical functions are orthogonal.P This explains why ′ ψi(x)= ψj(φ(x)) for some ji.

Example 29 Here is a diagram, created with a help of a computer, of Hecke equivalences between all the Gelfand pairs that produce matrices (A,B,C) (58) with n = 7:

(S ,PSL(3, Z )) (A , A ) (S ,S ) (Z , (Z ⋊Z ) ⋊Z ). 4 2 → 6 7 → 6 7 ← 3 7 3 2

Note that the graph is connected and all pairs are (strongly) Hecke equivalent to (S6,S7).

Example 30 We used a computer to generate the diagram of all Hecke equivalences of Gelfand pairs that correspond to matrix C

1 1 6  3 3 6  . −    4 4 0   −    25 The matrix is a member of a bigger family matrixes related to Sn Sk (n = 4, k = 2)and discussed ≀ in Section 3.2. We have a connected diagram of equivalences with the terminal group equal to S S (we omit precise formulas for inclusion H G): 4 ≀ 2 ⊂ A B C D E . . . . .

. . F G

A = A Z×4 ⋊Z ⋊Z 4 ⊂ 2 3 2 B = S ( Z×4 ⋊Z ⋊Z ) ⋊Z 4 ⊂ 2 3 2 2 C = (Z A ) ⋊Z (A×2 ⋊Z ) ⋊Z 3 × 4 2 ⊂ 4 2 2 D = (Z A ) ⋊Z (A×2) ⋊Z 3 × 4 2 ⊂ 4 4 E = S×2 (S×2) ⋊Z 4 ⊂ 4 2 F = A Z×4 ⋊Z ⋊Z 4 ⊂ 2 3 2 G = Z A (A×2) ⋊Z 3 × 4 ⊂ 4 2 Here is the diagram of maps of the dual family of groups. The pair which corresponds to S S 2 ≀ 4 from Section 3.2 is C′:

A′ B′ C′ . . .

. D′ E′ F ′ . .

26 A′ = S GL(2, Z ) 3 ⊂ 3 B′ = S (((Z D ) ⋊Z ) ⋊Z ) ⋊Z 4 ⊂ 2 × 8 2 3 2 C′ = Z S Z4 ⋊ S 2 × 4 ⊂ 2 4 D′ = Z A Z4 ⋊ A 2 × 4 ⊂ 2 4 E′ = A ((Z D ) ⋊Z ) ⋊Z 4 ⊂ 2 × 8 2 3 F ′ = S (((Z D ) ⋊Z ) ⋊Z ) ⋊Z 4 ⊂ 2 × 8 2 3 2 We see that the diagram of Hecke equivalences for the dual family is disconnected.

We come to the conclusion that the dual pair (H,ˇ Gˇ) for (H, G), even it exists, is not necessarily unique even if study pairs up to Hecke equivalence.

3 Examples

In this section we discuss some examples of the proposed duality. We will formulate a classical suﬃcient condition for (H, G) to be a Gelfand pair.

Proposition 31 Let ψ : G G be an involutive anti-automorphism such that ψ leaves each → double cosets HgH invariant. Then (H, G) is a Gelfand pair.

3.1 (Sn−1,Sn)

−1 Let Sn be a symmetric group on n letters (n 2,S := 1 ). The map σ σ leaves Sn ≥ 1 { } → −1 and Sn−1σn−1,nSn−1 invariant. So, by Proposition 31 (Sn−1,Sn) is a Gelfand pair. The set

X = Sn/Sn is isomorphic to 1,...,n with the natural Sn-action. X is a union of two −1 { } Sn -orbits 1,...,n 1 n . The regular Sn representation in C[X] is a direct sum of −1 { − } ∪ { } the trivial representation C and n 1-dimensional deﬁning representation. We conclude that − A = n(1,n 1),B = (1,n 1) and C is − − 1 n 1 Ln := − . (58)  n 1 (n 1)  − − −  

27 The formulas for A and B follow from equations (36) and (44). The formulas for A and B do not support nontrivial involutions π and µ. The formulas for C follows from Proposition 47 and Corollary 19.

The symmetry of the matrix C implies that (Sn−1,Sn) is a self-dual pair.

3.2 The group S S n ≀ k ×k A more complex example is the wreath product G = Sn Sk = S ⋊ Sk,n,k 2. It is a ≀ n ≥ symmetry group of a set of pairs

X = (i, j) 1 i n, 1 j k . { | ≤ ≤ ≤ ≤ }

×k An element g = (σ ,...,σk) ⋊ σ S ⋊ Sk acts by the formula 1 ∈ n

g(i, j) = (σl(i),σ(j)), l = σ(j).

×(k−1) The stabilizer of (1, 1) is the subgroup H = (Sn Sn ) ⋊ Sk . −1 × −1 The H-orbits of H in X are

o o o := (1, 1) (i, 1) 2 i n (i, j) 1 i n, 2 j k . 1 ∪ 2 ∪ 3 { } ∪ { | ≤ ≤ } ∪ { | ≤ ≤ ≤ ≤ }

Their respective cardinalities are 1,n 1,n(k 1). − − From this we conclude that G = H Hg H Hg H where ∪ 1 ∪ 2

×(k−1) ×(k) g = (σ , 1 ) ⋊ 1, g = (1 ) ⋊ σ , . 1 1 2 × 2 1 2

The anti-involution g g−1 leaves all three double cosets invariant and by Proposition 31 we → know that (H, G) is a Gelfand pair. The group G is compatible with projection p : X X = 1 ...,k ,p(i, j) = j. G action → 0 { } on the image of p factors through Sk. The pullback along p deﬁnes an inclusion C[X ] C[X]. 0 ⊂ C[X ] is a direct sum of the trivial C and k 1 dimensional representation T . The orthogonal 0 − complement Q to C[X ] in C[X] has dimension kn (1 + (k 1)) = (n 1)k. It must be an 0 − − − irreducible representation because (H, G) is a Gelfand pair and dim C[X X]G = 3. ×

28 Let us compute function Ψ2 corresponding to representation T . According to the formulas k k (40) and (43) this is done in terms of H-invariant normalized vector θ aiei ai = 0 . ∈ i=1 | i=1 In this formula ei is the standard basis for C[X0]. We choose it to benP P o { } k 1 θ = (k 1)e ei . − 1 − k(k 1) i ! − X=2 p The function (T (g)θ,θ), g Sk from (40) has the following values: ∈

1 if g Sk ∈ −1 (T (g)θ,θ)=  1  if g Sk σ Sk . − k−1 ∈ −1 12 −1  From this we immediately compute the values of Ψ2(x,x0), where x0 = (1, 1):

dim T = k−1 if x = (i, 1), 1 i n |X| nk ≤ ≤ Ψ2(x,x0)=   dim T = 1 if x = (i, j), 1 i n, 2 j k. − |X|(k−1) − nk ≤ ≤ ≤ ≤ From this we derive the following formulas:

(Xi, Ψ )= X Xi(x,x )Ψ (x,x )= 2 | | 0 2 0 x X X∈ k 1 , i = 1  − = (n 1)(k 1) , i = 2  − −  (k 1)n , i = 3. − −   These numbers constitute the second column of the matrix C. We recover the third column from equation (48). Here is the ﬁnal answer for C:

1 k 1 k(n 1) − − Mn,k =  n 1 (k 1)(n 1) k(n 1)  . (59) − − − − −    (k 1)n (1 k)n 0   − −   −1  As the double cosets are invariant with resect to g g , we conclude that Xi are invariant with → respect to π and µ. π and µ act trivially on Ψi because all the irreducible sub-representations in C[X] are real.

29 † We see that C corresponds to the column-wise action of Sk Sn on the set X. It is remarkable ≀ k n that duality doesn’t preserve cardinalities of the groups for Sn Sk = n! k! = k! n!= Sk Sn . | ≀ | 6 | ≀ |

3.3 Semidirect product with an abelian group

In this section we will find the dual pair for the pair (H, A ⋊ H). In this pair groups A and H are finite. A is an abelian group, equipped with an H-action. The embedding of H is defined by h (0, h). → There is an anti-involution ν on G defined by the formula (a, h) (h−1(a), h−1). It is the → composition of the inverse and the involution (a, h) ( a, h). ν preserves double cosets HaH. → − By Proposition 31 (H, A ⋊ H) is a Gelfand pair. Note that the homogeneous space X coincides with A. The group H is acting on X = A according to homomorphism φ : H Aut(A), which → defines the semi-direct product. The group A acts on X by translations. We use adjoint to φ to define the semidirect product Aˇ ⋊ H, where Aˇ is the Pontryagin dual Hom(A, C×). In this section we show that (H, Aˇ⋊ H) is dual to (H, A ⋊ H). Note that even though Aˇ⋊ H and A ⋊ H have the same cardinalities in general the groups are not isomorphic. The simplest × example is H = Fp acting on A = Fp-additive group of the finite field. The H orbits in X are the orbits of H-action on A. To simplify notations we fix a represen- tative ai oi A, χj oj Aˇ in each H-orbit. This way ∈ ⊂ ∈ ⊂ k k A = o(ai), Aˇ = o(χi). (60) i i G=1 G=1 An o(χi)-graded representation T of the group G is a representation that decomposes into the direct sum of linear spaces labelled by characters from a fixed G-orbit:

To(χi) = Tχ. χ∈Mo(χi)

The group G acts on T by isomorphisms g : Tχ Tgχ. The groups A acts on Tχ according to → the character χ.

30 Proposition 32 1. Any representation T of G has a decomposition

T = To(χi). (61) i M=1

2. To χ is G-irreducible iﬀ the graded component To χ is an irreducible representation of ( i) ( i)χi

St(χi).

3. C[X] splits into the direct sums (61) where T = 0 for each i. Moreover dimensions o(χi) 6 { }

of o(χi)-graded components Tχ,o(χi) are equal to one.

Proof. T decomposes into A-isotypic components by the characters χ Aˇ. The group G shuﬄes ∈ these components. The characters can be grouped according to orbits o(χi). By construction,

χ∈o(χi) Tχ is a subrepresentation in T . The proof follows from (60). ′ ′ L If T Tχ is a proper St(χi)-subrepresentation, then gT is a G subrepresentation χi ⊂ i g∈G χi of To(χi). Hence To(χi) is not irreducible. Conversely, any irreducibleP G- subrepresentation P of

T is generated by Pχ Tχ . o(χi) i ⊂ i As a representation of the group A, C[X] is isomorphic to C[A]. From the Fourier analysis on A, we know that its A-isotypic components are one-dimensional. Hence dim C[X]χ = 1 for all χ. The second statement follows from this.

Our next goal is to ﬁnd a formula for the functions Φi. To do this we have to ﬁnd a G-invariant vector θ T . ∈ o(χi)

Proposition 33 θ(a)= χ(a) (62) χ∈Xo(χi) is an H-invariant function on X.

Proof. Let θ(a)= θχχ(a) T C[X] be an H-invariant function. θ satisﬁes χ∈o(χi) ∈ o(χi) ⊂ P −1 −1 θ((0, g )a)= θχχ(g a)) = θχχ(a)= θ(a). χ∈Xo(χi) χ∈Xo(χi)

It implies that θg−1χ = θχ.

31 The function 1 θo(χi)(a)= χ(a) ˇ A o(χi) χ∈o(χi) | || | X q represents an H-invariant unit vector in To(χi). After a little algebra we ﬁnd that

1 1 (θi, Ti(l, g)θi)= χ(l), ψi(l, g)= χ(l) o(χi) A | | χ∈XO(χi) | | χ∈Xo(χi) The triple (A,B,C) in our case is

(Xi, Xj)= δij A o(ai) = δijAi, | || |

(Ψi, Ψj)= δij o(χi) = δijBi, | | (63) −1 (Xi, Ψj)= χ (a)= Cij. χ∈Xo(χj ), a∈o(ai)

Equations (63) make it obvious that a possible dual pair to H A ⋊ H is H Aˇ ⋊ H. ⊂ ⊂

3.4 Non examples

There are some classical examples of Gelfand pairs that do not have any dual counterparts. One set of such non examples is formed by Sk Sn k Sn (See [17] for classiﬁcation of multiplicity × − ⊂ free subgroups in Sn.) Computations with GAP4 (we omit details) show that Cn derived from

S Sn Sn is 2 × −2 ⊂ 1 n + 1 1 (n 1)(n + 2) 2 − Cn =  2n (n 2)(n + 1) (n 1)(n + 2)  . − − −  1 1   (n 1)n (n 1)(n + 1) (n 1)(n + 2)   2 − − − 2 −    Cn fails integrality test (56) for n 3. ≥ Gelfand pairs (G, G G) described in Remark 20 almost never pass integrality tests (55,56). × Rare exceptions from the above rule come some solvable G. The nature of these groups deserve a closer study.

32 3.5 On classiﬁcation of dual pairs

In this section we present our attempt to compile a list of matrices C that come from dual Gelfand pairs. The computation of matrix C, if done by hand, could be very time-consuming. This is why we used the GAP4 system to accelerate the calculations. By (52) duality doesn’t change X . GAP4 has a library of all transitive eﬀective actions on sets with cardinality 30. | | ≤ Given a group and a transitive action on X we recover H without troubles, as a stabilizer of a point. This way we can systematically search for dual pairs. The data that must be present in an ideal list might take some space. In such a list each matrix C should be accompanied by the description of the generators of G S . The list should also contain a description of all ⊂ |X| Hecke equivalences φ : G G′ in the case C happen to correspond to more then one pair. It → is clear that there is a room in the list for more interesting details. To save space I decided to drastically limit myself with a description only of C and with a record of cardinality r(C) of the set (H, G) C(H, G) = C of the isomorphism classes of pairs. I will also omit matrices that { | } break into nontrivial tensor products.

Let us fix the notations used in the table. Matrix Ln is defined in (58). Matrix En is ij (ζn ), i, j = 0,...n 1. ζn is a primitive root of unity of order n. Matrix Mn,k is defined in (59). − In the table below we will be writing r(C) in the exponent of C. Double entries in the second

33 column correspond to dual pairs. Single entries correspond to self-dual pairs:

X C OutH (G), Gal(L/Q) | | 2 L1 1 , 1 2 { } { } L1 1 , 1 3 3 { } { } 1 E3 Z2, Z2 L2 1 , 1 4 { } { } 4 M 1 1 , 1 2,2 { } { } 1 E4 Z2, Z2 3 L5 1 , 1 { } {1 } 1 2 2  3 2 4  5 2 2ζ5 + 2ζ5 2ζ5 + 2ζ5 Z2, Z2  4 3 2   2 2ζ + 2ζ5 2ζ + 2ζ   5 5 5    1 E5 Z4, Z4 L4 1 , 1 6 { } { } 3 3 M2,3 1 , 1 M3,2 1 , 1 6 2 { } { } { } {1 } 1 1 1 3 1 1 2 2  1 1 1 3   1 1 2ζ2 2ζ  − Z , Z 3 3 Z , Z  2  2 2  2  2 2  2 2ζ3 2ζ3 0   1 1 2ζ3 2ζ3       2     2 2ζ 2ζ3 0   3 3 0 0   3   −      4 L7 1 , 1 { } { 1} 1 3 3  6 5 3 4 2  3 3ζ7 + 3ζ7 + 3ζ7 3ζ7 + 3ζ7 + 3ζ7 Z2, Z2  4 2 6 5 3  ∗  3 3ζ + 3ζ + 3ζ7 3ζ + 3ζ + 3ζ   7 7 7 7 7    1 7 1 2 2 2  4 3 5 2 6  2 2ζ7 + 2ζ7 2ζ7 + 2ζ7 2ζ7 + 2ζ7 Z , Z  6 4 3 5 2  3 3  2 2ζ7 + 2ζ7 2ζ7 + 2ζ7 2ζ7 + 2ζ7     5 2 6 4 3   2 2ζ + 2ζ 2ζ + 2ζ7 2ζ + 2ζ   7 7 7 7 7    1 E7 Z6, Z6

· · · · · · The matrix marked by astrisque corresponds to a self-dual pair with a nontrivial self- ∗ duality isomorphism (26). The number of entries in the table grows fast with X (for X = 12 | | | | this number is about 30) and proposed presentation of data becomes impractical.

In the adopted notation matrix C corresponding to (S ,S S ), from example (1), is M , . 3 3 × 3 3 2 The matrix M2,3 corresponds to (Z4,S4).

34 3.6 Some sporadic examples

Mathieu group G = M11, the smallest in the sporadic family, acts transitively on the 22 objects

(see [4] p.221 for details). In this case H is the alternating group A6. The C matrix is

1 10 11  1 10 11  . −    20 20 0   −    The transposed matrix corresponds to the group Gˇ = (M M ) ⋊Z with the subgroup Hˇ 11 × 11 2 isomorphic to M S . The action of Gˇ comes from the action of M on Ω, Ω = 11. Let us 11 × 6 11 | | ′ ′ take two copies of Ω. Then M M acts on Ω Ω . The generator a Z acts by ωi ω . 11 × 11 ∈ 2 → i Matrix C doesn’t support nontrivial π and µ. ` There is a self-dual example with C equal to

1 11 11 55 66  11 121 11 55 66  − − −    11 11 121 55 66  .  − − −     55 55 55 385 330   − − −     66 66 66 330 396   − − −    It is related to the exceptional group G = M and a subgroup H = PSL (Z ) M M . 12 2 11 ⊂ 11 ⊂ 12 Involutions π and µ also act trivially. For more on multiplicity free subgroups in sporadic groups see [1].

4 Some open questions

In this section I collected several questions related to the subject of the paper that are worth of a closer look.

4.1 On the relation to other dualities

The ﬁrst question that needs to be addressed is rather vague: what is the relation of the proposed duality to other dualities known in mathematics: Mirror symmetry and Langlands

35 duality. Though, the constructions in this paper were partly motivated by Satake isomorphism, the relation to main body of Langlands conjectures is not clear. The spaces L(N, M) (or better L(N,N ′, M) = γ : [0, 1] M γ(0) N, γ(1) N ′ ) where N,N ′ are Lagrangian submanifold { → | ∈ ∈ } in symplectic M, play an important role in the deﬁnition of Floer homology and Fukaya category. This might give an indication of a possible relation of proposed duality to Mirror symmetry.

4.2 Duality for Gelfand pairs of Lie groups and locally compact groups

In order to define a Gelfand pair in the context of differential geometry we consider the algebra of differential operators Diff(X) on the homogeneous space X = G/H (see [16] for more details). The algebra Diff(X)G in general is noncommutative. When it is commutative H, G is by definition a Gelfand pair. Diff(X)G is analogous to (C[X X]G, ). A possible candidate for × × the role of (C[X X]G, ) is the space of G-invariant functions on the cotangent bundle T ∗X × · with point-wise multiplication. It is not clear if Diff(X)G and C∞(T ∗X) support any traces, which are necessary to put them in our framework. Modulo these details one can ask a question of constructing a dual pair H,ˇ Gˇ, where the role Diff(X)G and C∞(T ∗X) is interchanged. It is not clear at present if this duality can be illustrated by any interesting examples. The simplest question that waits its answer is what is the dual pair for S1 SU and whether the dual pair ⊂ 2 exists at all. Perhaps, we have to extend the area of search to include locally compact dual pairs. The fundamental group, which was used in Example 3, is a part of a package that contains also the higher homotopy groups. The higher homotopy groups form a graded Lie algebra with respect to Whitehead product. In fact the theory of rational homotopy type of Sullivan tells us that the the homotopy category of rational differential graded Lie algebras with nilpotent finite type homology is equivalent to the rational homotopy category of nilpotent topological spaces with finite type rational homology. Would it be possible to combine this equivalence and the Lie algebra technique used in the differential geometry to construct interesting examples of topological Gelfand pairs?

36 4.3 On the categoriﬁcation

G Objects of the category V ect[X X] [7] are ﬁnite-dimensional linear spaces Vx,y,x,y X × ∈ together with a ﬁnite group action. R(g) : Vx,y Vgx,gy, g G, X. There is a map from → ∈ G G G K-functor of V ect[X X] to C[X X] Vx,y dim Vx,y. V ect[X X] has two tensor × × → × products:

(V W )x,y = Vx,y Wx,y (V ⊠ W )x,y = Vx,z Wz,y. ⊗ ⊗ z ⊗ M Involutions act by Π(V )x,y = V x,y, M(V )x,y = V y,x. If H, G is a Gelfand pair it is possible to choose a tensor subcategory F [X X]G in V ect[X X]G in such a way that dim : K(F [X X]G) × × × ⊗ C C[X X]G is an isomorphism. Is it true that V ect[X X]G, or, perhaps, better F [X X]G, → × × × is a symmetric monoidal category? What is the formula for commutativity morphisms? Is it possible to categorify duality? Naively, categorification should take a form of a tensor functor D : F [X X]G F [Xˇ Xˇ]Gˇ , which interchanges tensor structures and involutions Π and M. On the × → × level of K-functors it should define an isomorphism C[X X]G C[Xˇ Xˇ]Gˇ that interchanges × → × the algebra structures. The problem has no solution the way it is formulated. The simplest counterexample is G = Z ,H = 1 . We know that vector spaces give us categorification of 5 { } integers. In order to accommodate entries of matrix C in the categorical language we have to categorify the ring of integers of the cyclotomic field and use it to properly modify F [X X]G. × Still, even in the case that C has integral coefficients (e.g. in case of Sn−1,Sn) categorification is unknown. Such a categorification, if it exist, would resemble a construction from [12]. Note that formulas (39) bases Xi and Φj have analogues in the Geometric Langlands theory. Element { } { } Xi is similar to the image in the K-theory of a skyscraper sheaf with support on the strata of the affine Grassmannian. Φj is analogous to the image of a perverse sheaf. The main distinction from the Geometric Langlands theory is that our matrix C is never upper-triangular.

4.4 On the suﬃcient conditions for existence of the dual pair

The author is not aware of any such a condition except a condition to be a ﬁnite abelian group. Experimental data (about hundred examples) doesn’t contradict the following conjecture: If the

37 matrix C corresponding to Gelfand pair H, G satisfying (55) and (56) then there is a dual pair H,ˇ Gˇ.

4.5 On the String Topology

Not too many examples of topological Gelfand pairs discussed in the introduction are known. The most basic is pt S2n+1. H (N, M), H∗(N, M) in this case are polynomial algebras in { } ⊂ ∗ one variable of degree 2n. Simply-connected compact Lie groups give another class of examples. Again, N = 1 is one-point set. A product of such manifolds, we denote it by P , can be { } interpreted as a topological analogue of a finite abelian group (it is also a self-dual object). Let us consider a fiber bundle M with a base N and a fiber P . Let us also assume that the bundle has a section σ : N M. This structure is analogous to a semidirect product A ⋊ H from → Section 3.3. We conjecture that (N, M) is a topological Gelfand pair. Under what conditions it is self-dual? The author is planning to address these questions in the following publications.

References

[1] T. Breuer and K. Lux. The multiplicity-free permutation characters of the sporadic simple groups and their automorphism groups. Communications in Algebra, 24(7):2293–2316, 1996.

[2] R. Brown. Topology and Groupoids. BookSurge Publishing, 2006.

[3] M. Chas and D. P. Sullivan. String topology. arXiv.org/abs/math/9911159.

[4] J.H. Conway. Three lectures on exceptional groups. In M.B. Powell and G. Higman, editors, Finite simple groups, pages 215–247. Academic Press, 1971.

[5] C. W. Curtis and I. Reiner. Representation Theory Of Finite Groups And Associative Algebras. Interscience Publishers, 1962.

[6] E. S. Egge. A generalization of the Terwilliger Algebra. Journal of Algebra, 233(1):213–252, 1 November 2000.

38 [7] P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik. Tensor Categories, volume 205 of Mathematical Surveys and Monographs. AMS, 2015.

[8] E. Frenkel. Lectures on the Langlands Program and Conformal Field Theory. In P. Cartier, P. Moussa, B. Julia, and P. Vanhove, editors, Frontiers in Number Theory, Physics, and Geometry II, pages 387–533. Springer, Berlin, Heidelberg, 2007.

[9] N. Hingston and A. Oancea. The space of paths in complex projective space with real boundary conditions. arXiv:1311.7292 [math.GT].

[10] A. A. Kirillov. Representation Theory and Noncommutative Harmonic Analysis I, volume 22 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin Heidelberg GmbH, 1994.

[11] E. Lupercio, B. Uribe, and M. A Xicotencatl. Orbifold string topology. Geometry and Topology, 12:2203–2247, 2008.

[12] I. Mirkovi´cand K. Vilonen. Geometric Langlands duality and representations of algebraic groups over commutative rings. Annals Of Mathematics, 166(1):95–143, 2007.

[13] J. Neukirch. Algebraic Number Theory. Springer, 1999.

[14] L.D. Pontryagin. Topological groups. Princeton Univ. Press, 1958.

[15] D. Sullivan. Open and closed string ﬁeld theory interpreted in classical algebraic topology. In Topology, geometry and quantum ﬁeld theory, volume 308 of London Math. Soc. Lecture Note Ser., pages 344–357. Cambridge Univ. Press, Cambridge, 2004.

[16] E.B. Vinberg. Commutative homogeneous spaces and coisotropic actions. UMN, 56(1):3–62, 2001.

[17] M. Wildon. Multiplicity-free representations of symmetric groups. Journal of Pure and Applied Algebra, 213:1464–1477, 2009.